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Towards a critical endpoint in the valence fluctuating Eu(Rh$_{1-x}$Co$_{x}$)$_2$Si$_2$ system

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Pith's one-line read The critical endpoint of the valence transition in Eu(Rh$_{1-x}$Co$_x$)$_2$Si$_2$ lies between x = 0.119 and x = 0.166, where the first-order jump gives way to a continuous valence crossover.

desk verdict Single-crystal Co series gives a plausible CEP near x~0.14, but the upper bound of the claimed window rests on an unshown measurement. read the letter →

arxiv 2505.01127 v1 pith:SV5LAO42 submitted 2025-05-02 cond-mat.str-el

Franziska Walther , Michelle Ocker , Alexej Kraiker , Nubia Caroca-Canales , Silvia Seiro , Kristin Kliemt , Cornelius Krellner This is my paper · ORCID
classification cond-mat.str-el
keywords valencetransitioncriticalendpointEu(Rh1-xCox)2Si2EuRh2Si2chemicalpressurethermalhysteresiscrossoversingle-crystalgrowth
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reports single-crystal growth of the series Eu(Rh$_{1-x}$Co$_x$)$_2$Si$_2$ and uses cobalt substitution as chemical pressure to sweep europium's valence instability through its end point. The central claim is that the first-order valence transition—where Eu switches from a magnetic divalent to a less magnetic intermediate-valence state—survives only up to a cobalt content between $x = 0.119$ and $x = 0.166$, and that above this window the transition smooths into a continuous valence crossover. That location matters because, compared with the iridium-substituted analogue, reaching the critical endpoint here requires far less substituent, hence less chemical disorder, making the system a cleaner platform for studying the proposed critical elasticity of the lattice near such endpoints. The paper also maps several antiferromagnetic phases and a basal-plane magnetic anisotropy in the pure compound.

What carries the argument

The load-bearing control parameter is chemical pressure generated by substituting smaller Co for Rh in the ThCr$_2$Si$_2$ structure: the unit-cell volume shrinks with $x$, compressing the Eu site and destabilizing the large-volume Eu$^{2+}$ state. The observable that carries the argument is thermal hysteresis—the gap between the valence transition temperatures measured on cooling and heating—whose width tracks the first-order character of the transition. The paper uses the collapse of that hysteresis to locate the critical endpoint, and converts $x$ to an equivalent pressure through $\Delta V/V \cdot K$ with a bulk modulus $K = 100$ GPa, which permits direct comparison with hydrostatic-pressure experiments.

What would settle it

Grow a single crystal with composition near $x=0.15$, determine the cobalt content by an independent method such as X-ray absorption or wavelength-dispersive spectroscopy, and measure temperature-dependent magnetization and heat capacity on the same crystal: a thermal hysteresis appearing for $x>0.166$, or its absence for $x<0.119$, would disprove the claimed window. Re-measuring the heat capacity and Eu valence of the $x_{\rm EDX}=0.166$ sample would directly test the unshown data point.

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Extended reading notes

Core claim

The paper establishes that in Eu(Rh$_{1-x}$Co$_x$)$_2$Si$_2$ a sharp temperature-induced first-order valence transition is present for $0.081 \leq x_{\rm EDX} \leq 0.119$, signalled by large thermal hysteresis in magnetization (about 18 K at $x=0.081$), resistivity (up to 23 K), and heat capacity (about 10 K at $x=0.126$). As $x$ increases, the transition temperature rises almost linearly from about 37 K to about 105 K and the hysteresis narrows, indicating approach to a critical endpoint. From the disappearance of hysteresis between $x_{\rm EDX}=0.119$ and $x_{\rm EDX}=0.166$, the paper places the critical endpoint in the window $0.119 < x_{\rm EDX} < 0.166$; the upper bound rests on heat-capacity data from a polycrystalline sample that are not shown. Above this regime, for $x_{\rm EDX}\approx 0.23$, only a broad valence crossover near 220 K remains, with no first-order character. The authors conclude this is the same valence instability seen under hydrostatic pressure on EuRh$_2$Si$_2$, with the endpoint reached at an estimated chemical pressure of about 1.7 GPa.

Load-bearing premise

The upper edge of the claimed critical-endpoint window rests on one unshown heat-capacity measurement of a polycrystalline sample with $x_{\rm EDX}=0.166$; if that sample's cobalt concentration or its classification as a crossover is wrong, the endpoint's upper bound shifts.

Editorial extensions

If this is right

  • Within the first-order regime, $T_V$ rises almost linearly from about 37 K at $x_{\rm EDX}=0.081$ to about 105 K at $x_{\rm EDX}=0.119$, so composition acts as a clean dial for the transition temperature.
  • The disappearance of hysteresis between $x=0.119$ and $0.166$ marks a boundary: below it, cooling and heating trace different states; above it, the valence change is continuous.
  • The equivalent hydrostatic pressure at the endpoint is approximately 1.7 GPa, close to the 2.05 GPa estimate for pure EuRh$_2$Si$_2$, so chemical and external pressure drive the same valence instability.
  • Because the endpoint is reached by substituting only about 12–16% cobalt, the chemical disorder at the critical point should be substantially smaller than in Eu(Rh$_{1-x}$Ir$_x$)$_2$Si$_2$, whose endpoint lies near $x \simeq 0.5$–$0.75$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A sharper test of the window would be to extrapolate the measured hysteresis width to zero against $x$; the paper does not perform that extrapolation, but its own data make it feasible.
  • If the endpoint is a genuine thermodynamic critical point, the elastic response—for instance the bulk modulus—should show anomalies in the same concentration range; the paper's motivation suggests this is the intended next measurement.
  • The overlap of the Co-K and Eu-L$\gamma_1$ lines in EDX could bias $x_{\rm EDX}$ downward, so the true endpoint composition may differ systematically from the reported window; valence-sensitive spectroscopy on the endpoint samples would settle this.
  • A possible fifth magnetic phase at low fields in pure EuRh$_2$Si$_2$ is only suspected from the data; field-angle-resolved magnetization could test whether a fan or spiral ground state actually lies below $B^*$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript reports single-crystal growth of EuRh2Si2 and Eu(Rh1−xCox)2Si2 (x ≤ 0.23) by flux methods, and uses magnetization, resistivity, and heat-capacity measurements to map the temperature–substitution phase diagram. For EuRh2Si2 the authors identify several antiferromagnetic phases and a weak in-plane anisotropy between [100] and [110]. For substituted samples, they observe a sharp first-order valence transition with pronounced thermal hysteresis for x_EDX between 0.081 and 0.126, and locate the critical endpoint (CEP) of this transition in the range 0.119 < x_EDX < 0.166. Above this range they identify a valence-crossover regime. The central claim is that this substitution series reaches the CEP at a lower substituent concentration than Eu(Rh1−xIrx)2Si2, making it a promising system for studying critical elasticity with reduced disorder.

Significance. If the CEP localization is correct, the paper delivers a materially useful result: a substitution series that approaches the critical endpoint of a first-order valence transition with relatively low chemical disorder, and a consistent set of first-order signatures (hysteresis in susceptibility, resistivity, and heat capacity) on single crystals. The detailed B–T phase diagrams for EuRh2Si2, including in-plane anisotropy, also provide a solid experimental foundation for this compound. The paper's own pressure-equivalence estimate (using K = 100 GPa and Vegard's law) is clearly presented as a conversion, not as a derivation, so the central qualitative conclusion does not depend on circular reasoning.

major comments (3)
  1. [Section III.C.4 and Fig. 7] The upper bound of the CEP window, x_EDX < 0.166, rests entirely on the statement that a polycrystalline sample with x_EDX = 0.166 'enters the valence crossover regime' based on heat-capacity data '(not shown)'. Since the CEP is identified by the disappearance of thermal hysteresis, the negative observation at this composition is load-bearing. The paper must show the C_p(T) cooling/heating data for this sample, along with its EDX-determined composition and a statement of how the crossover was assigned. As written, the reader cannot verify that the anomaly is non-hysteretic rather than absent, broadened, or obscured by the acknowledged composition uncertainties.
  2. [Table I and Section III.C] The compositions quoted in the text (x_EDX = 0.081, 0.116, 0.119, 0.126, 0.166, 0.228, 0.229) do not match the batch-averaged values in Table I (0.06, 0.08, 0.11, 0.12, 0.23). If the text values are obtained from the specific single crystal measured, that should be stated explicitly for each sample, and the x_EDX = 0.166 polycrystal should be included in the growth table. Without this connection, the phase diagram in Fig. 7 cannot be reproduced from the reported growths.
  3. [Section III.C.4 and Fig. 1(c)] The phase diagram in Fig. 7 includes data points for x_EDX = 0.166 and x_EDX = 0.23, but the unit-cell volume data in Fig. 1(c) switch between single crystals and polycrystalline samples without a clear distinction in the main text. Since the pressure conversion pCo = ΔV/V·K depends on these volume data, the provenance of each volume point used for the pressure comparison should be stated, and the uncertainty in K (taken as a 'typical' 100 GPa from Ref. [22]) should be propagated into the reported pressure estimates.
minor comments (5)
  1. [Abstract] The word 'substitition' should be 'substitution'.
  2. [Section III.C.3] The heat-capacity hysteresis is reported as a 'large difference of ∼10 K' between cooling (96 K) and heating (106 K), but the resistivity hysteresis for the same nominal composition (x_EDX = 0.126) is given as 15 K; the discrepancy between these values and their possible origin (different sweep rates, thermal lags) deserves a brief comment.
  3. [Section III.C.1] The sample labels in Fig. 5(a) include both x_EDX = 0.119 and x_EDX = 0.116, and the text states that 'slight changes in the Co substitution induce large differences in TV'; given the EDX Co-K/Eu-Lγ1 overlap, the authors should indicate how robust the 0.116 vs 0.119 distinction is given the stated EDX uncertainty.
  4. [Section III.C.4] The sentence 'with the intermediate Eu (2+δ)+ state at high temperatures' is incomplete as written; it should describe what is observed (e.g., a broad anomaly in C_p or χ) for the x_EDX = 0.166 polycrystal.
  5. [General] The text uses both 'a−a plane' and 'a−a plane' without italicizing the crystallographic axes consistently; please standardize notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the CEP localization is an empirical interpolation from direct hysteresis measurements, not a fitted or self-referential derivation.

full rationale

The central claim, localization of the critical endpoint to 0.119 < x_EDX < 0.166, is an empirical interpolation from direct observations: thermal hysteresis in magnetic susceptibility (Fig. 5a), resistivity (Fig. 5b), and heat capacity (Fig. 6) for x_EDX up to 0.126, plus the stated absence of hysteresis and crossover behavior for x_EDX = 0.166 (heat-capacity data not shown) and for x_EDX = 0.228/0.229. No parameter is fitted to these data and then used to predict the same quantity; the endpoint is not derived from the pressure-conversion formula. The p_Co = (ΔV/V)·K conversion uses external inputs (Vegard's law and K = 100 GPa from cited literature) and is used only for comparison with external hydrostatic-pressure experiments by Honda et al., not to locate the endpoint. The self-citations that appear (e.g., refs. 14, 25, and 33, involving co-author S. Seiro or the Frankfurt group) support background comparisons, such as pure EuRh2Si2 magnetization, the Ir-substitution phase diagram, and composition sensitivity; they are not load-bearing for the new CEP claim. The 'not shown' heat-capacity evidence for x_EDX = 0.166 is a missing-data and reproducibility concern, not a circularity: if the sample composition or the assignment of the crossover regime were wrong, the upper bound would shift, but the argument does not reduce to its own inputs by construction. No circular step is therefore identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim is experimental; no parameter is fitted to define the CEP. The paper relies on literature values (bulk modulus K=100 GPa, Vegard's law) to convert chemical to hydrostatic pressure, and on prior XAS and photoemission to assign the transition to a Eu valence change. The CEP location itself is determined directly from measured hysteresis, so circularity burden is low.

assumptions (4)
  • domain assumption A typical bulk modulus of K=100 GPa for EuRh2Si2, taken from Ref [22], is used to convert chemical substitution into pressure via pCo = DeltaV/V * K.
    Used in Sections I and III.C.4 to compare the CEP pressure with hydrostatic-pressure experiments. The value is literature-based, not measured here.
  • domain assumption Unit-cell volume follows Vegard's law with a linear decrease at low Co concentrations, with a stronger decrease for x > 0.17.
    Used in Section I to predict the substitution range for the valence transition and in Fig. 1c to derive chemical pressure from the volume change.
  • domain assumption The observed first-order transition in magnetization, resistivity, and heat capacity corresponds to a Eu valence transition.
    The paper does not measure the Eu valence on these single crystals; it relies on prior XAS and photoemission studies on polycrystals (Refs 26 and 27) and the standard signatures of valence transitions in Eu 122 compounds (Ref 22).
  • domain assumption Co substitution acts like hydrostatic pressure with only minor additional disorder effects.
    Section III.C.4 concludes that the agreement with Honda's pressure data shows disorder has minor effects; this equivalence is assumed rather than directly demonstrated.

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Pith. "Pith review of Towards a critical endpoint in the valence fluctuating Eu(Rh$_{1-x}$Co$_{x}$)$_2$Si$_2$ system." pith.science (2026). https://pith.science/paper/SV5LAO42

@misc{pith2026250501127,
  author       = {Pith},
  title        = {Pith review of: Towards a critical endpoint in the valence fluctuating Eu(Rh$_1-x$Co$_x$)$_2$Si$_2$ system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SV5LAO42}},
  note         = {Machine review of arXiv:2505.01127}
}
abstract

We report on the successful single crystal growth of pure EuRh${_2}$Si${_2}$ and of Eu(Rh$_{1-x}$Co$_{x}$)$_2$Si$_2$ with $x\leq0.23$ by the flux method. Through Co substitution, EuRh$_2$Si$_2$ can be tuned from stable antiferromagnetism via a valence-transition state towards the valence-crossover regime. From magnetization measurements, we constructed a $B - T$ phase diagram for EuRh${_2}$Si${_2}$ comprising multiple magnetic phases and showing a sizable magnetic anisotropy within the basal plane of the tetragonal unit cell. This indicates a complex antiferromagnetic ground state for $x=0$. By applying positive chemical pressure through the substitution series Eu(Rh$_{1-x}$Co$_{x}$)$_2$Si$_2$, a sharp temperature-induced first-order phase transition is observed in magnetization, resistivity and heat capacity for 0.081 $\leq$ $x$ $\leq$ 0.119. The critical end point of this valence transition is located in the phase diagram in the vicinity of 0.119 $<x_{\rm EDX}<$ 0.166. At higher substitution level, the system reaches a valence-crossover regime. The obtained results are presented in a temperature-substitition phase diagram.

Figures

Figures reproduced from arXiv: 2505.01127 by the authors.

Figure 1
Figure 1. FIG. 1. (a): ThCr [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. EuRh [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Eu(Rh [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (4 more)
Figure 3
Figure 3. Figure 3: FIG. 3. EuRh [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Eu(Rh [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Eu(Rh [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Temperature - substitution phase diagram for [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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Reference graph

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